Area Proportional Drawings of Intersecting Families of Simple Closed Curves

Abstract:

A FISC, or family of intersecting simple closed curves,
is a collection of simple closed curves in the plane with
the properties that there is some open region common
to the interiors of all the curves, and that every two
curves intersect in finitely many points or arcs.
Let F be a FISC with a set of open regions R. F
is said to be area-proportional with respect to weight
function w : R --> R+ if there is a positive constant
C such that for any two finite regions, r1 and r2,
area(1)/area(2) = C w(1)/w(2).
We consider F as a directed plane graph, G(F), where
the curve intersections are vertices and the curve arcs
between vertices are edges. Edges are directed so that
each of F's curves is traversed in a clockwise fashion.
The directed plane dual of G(F), denoted D(F), has
edges oriented to indicate inclusion in fewer interiors of
the curves. The graph G(F) has an area-proportional
drawing with respect to w if there is some FISC C that
is area-proportional to w and where F can be transformed
into C by a continuous transformation of the
plane. We describe an O(n|V|) algorithm for creating
an area-proportional drawing of G(F) = (V,E) where F
is a FISC with n curves and D(F) has only one source
and only one sink. For the case of n-Venn diagrams,
since |V| <= 2n-2, this yields an
O(|V| lg|V|) drawing
algorithm.

Stirling Chow,
Department of Computer Science,
University of Victoria, Canada.
Frank Ruskey,
Department of Computer Science,
University of Victoria, Canada.